Group #49091089 — subgroups

Subgroups — #1564×C₂₈₁ #49091089

Order 35968.

180 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×C₂₈₁	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₈₁	C₂	1	2	yes	maximal, complemented
4	#334×C₂₈₁	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₂₈₁	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₈₁	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₂₈₁	C₂²	1	4	yes	—
9	C₂²⋊C₈×C₂₈₁	—	2	2	no	—
10	C₄×C₈×C₂₈₁	—	2	2	no	—
11	C₄×C₈×C₂₈₁	C₄	1	1	yes	characteristic
12	C₄×C₈×C₂₈₁	C₄	1	1	yes	characteristic
13	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₈₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₈₁	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₂₈₁	—	2	2	no	—
19	C₂×C₈×C₂₈₁	—	2	4	no	—
20	C₂×C₈×C₂₈₁	C₈	1	4	yes	complemented
21	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₂₈₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₈₁	D₄	1	1	yes	characteristic
27	C₄²×C₂₈₁	Q₈	1	1	yes	characteristic
28	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
29	C₂³×C₂₈₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₈₁	—	2	2	no	—
31	C₂×C₄×C₂₈₁	—	2	2	no	—
32	C₂×C₄×C₂₈₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₈₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₈₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₈₁	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₂₈₁	M₁₆	1	2	yes	—
37	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₂₈₁	—	2	1	no	—
41	C₈×C₂₈₁	—	2	1	no	—
42	C₈×C₂₈₁	—	4	4	no	—
43	C₈×C₂₈₁	C₂×C₈	1	2	yes	—
44	C₈×C₂₈₁	C₂×C₈	1	2	yes	—
45	C₂²×C₂₈₁	—	2	1	no	—
46	C₂²×C₂₈₁	—	2	1	no	—
47	C₂²×C₂₈₁	C₄⋊C₈	1	2	yes	—
48	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₂₈₁	—	2	1	no	—
50	C₄×C₂₈₁	—	2	1	no	—
51	C₄×C₂₈₁	C₄⋊C₈	1	2	yes	—
52	C₄×C₂₈₁	#108	1	2	yes	—
53	C₄×C₂₈₁	C₂²×C₈	1	1	yes	characteristic
54	C₄×C₂₈₁	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₂₈₁	#135	1	2	yes	—
56	C₂×C₂₈₁	—	2	2	no	—
57	C₂×C₂₈₁	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₂₈₁	#427	1	1	yes	characteristic
59	C₂×C₂₈₁	C₂×#108	1	1	yes	characteristic
60	C₂₈₁	#1564	1	1	yes	characteristic, direct product factor, Sylow
61	#1564	C₂₈₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	#431	C₂×C₂₈₁	1	2	yes	—
63	C₂×C₄×C₈	C₂×C₂₈₁	1	1	yes	characteristic
64	#335	C₂×C₂₈₁	1	2	yes	complemented
65	#334	C₂×C₂₈₁	1	2	yes	complemented
66	C₂²×C₈	C₄×C₂₈₁	1	1	yes	characteristic
67	C₂²×C₈	C₄×C₂₈₁	1	1	yes	characteristic
68	C₂×C₄²	C₂²×C₂₈₁	1	1	yes	characteristic
69	C₄⋊C₈	C₂²×C₂₈₁	1	4	yes	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	C₄×C₂₈₁	1	1	yes	characteristic
73	C₄×C₈	C₄×C₂₈₁	1	1	yes	characteristic
74	C₄×C₈	C₂²×C₂₈₁	1	1	yes	characteristic
75	C₄×C₈	C₂²×C₂₈₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄×C₂₈₁	1	1	yes	characteristic
77	C₂²×C₄	D₄×C₂₈₁	1	1	yes	characteristic
78	C₂²×C₄	Q₈×C₂₈₁	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₈×C₂₈₁	1	4	yes	complemented
82	C₂×C₈	C₂×C₄×C₂₈₁	1	1	yes	characteristic
83	C₂×C₈	C₂×C₄×C₂₈₁	1	1	yes	characteristic
84	C₂×C₈	C₂×C₄×C₂₈₁	1	1	yes	characteristic
85	C₂×C₈	C₂×C₄×C₂₈₁	1	1	yes	characteristic
86	C₄²	C₂×C₄×C₂₈₁	1	1	yes	characteristic
87	C₄²	D₄×C₂₈₁	1	1	yes	characteristic
88	C₄²	Q₈×C₂₈₁	1	1	yes	characteristic
89	C₄²	C₂³×C₂₈₁	1	1	yes	characteristic
90	C₂³	C₄⋊C₄×C₂₈₁	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₄⋊C₄×C₂₈₁	1	1	yes	characteristic
94	C₂×C₄	C₄⋊C₄×C₂₈₁	1	1	yes	characteristic
95	C₂×C₄	C₄⋊C₄×C₂₈₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₈×C₂₈₁	1	2	yes	—
97	C₂×C₄	M₁₆×C₂₈₁	1	2	yes	—
98	C₂×C₄	C₂²×C₄×C₂₈₁	1	1	yes	characteristic
99	C₂×C₄	C₂×D₄×C₂₈₁	1	1	yes	characteristic
100	C₂×C₄	C₂×Q₈×C₂₈₁	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	4	4	no	—
104	C₈	C₂×C₈×C₂₈₁	1	2	yes	—
105	C₈	C₂×C₈×C₂₈₁	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₄⋊C₈×C₂₈₁	1	2	yes	—
109	C₂²	C₂×C₄⋊C₄×C₂₈₁	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	C₄⋊C₈×C₂₈₁	1	2	yes	—
113	C₄	#108×C₂₈₁	1	2	yes	—
114	C₄	C₂²×C₈×C₂₈₁	1	1	yes	characteristic
115	C₄	C₂×M₁₆×C₂₈₁	1	1	yes	characteristic
116	C₄	#135×C₂₈₁	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	C₂×C₄⋊C₈×C₂₈₁	1	1	yes	characteristic
119	C₂	#427×C₂₈₁	1	1	yes	characteristic
120	C₂	C₂×#108×C₂₈₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#431×C₂₈₁

C₂

yes

maximal

C₂×C₄×C₈×C₂₈₁

C₂

yes

maximal, characteristic

#335×C₂₈₁

C₂

yes

maximal, complemented

#334×C₂₈₁

C₂

yes

maximal, complemented

C₂²×C₈×C₂₈₁

C₄

yes

characteristic

C₂²×C₈×C₂₈₁

C₄

yes

characteristic

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₈₁

C₂²

yes

—

C₂²⋊C₈×C₂₈₁

—

C₄×C₈×C₂₈₁

—

C₄×C₈×C₂₈₁

C₄

yes

characteristic

C₄×C₈×C₂₈₁

C₄

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

D₄

yes

characteristic

C₂²×C₄×C₂₈₁

Q₈

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

C₈

yes

complemented

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₄²×C₂₈₁

C₂×C₄

yes

characteristic

C₄²×C₂₈₁

D₄

yes

characteristic

C₄²×C₂₈₁

Q₈

yes

characteristic

C₄²×C₂₈₁

C₂³

yes

characteristic

C₂³×C₂₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×C₈

yes

—

C₂×C₄×C₂₈₁

M₁₆

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₈×C₂₈₁

—

C₈×C₂₈₁

—

C₈×C₂₈₁

—

C₈×C₂₈₁

C₂×C₈

yes

—

C₈×C₂₈₁

C₂×C₈

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₄⋊C₈

yes

—

C₂²×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₄⋊C₈

yes

—

C₄×C₂₈₁

#108

yes

—

C₄×C₂₈₁

C₂²×C₈

yes

characteristic

C₄×C₂₈₁

C₂×M₁₆

yes

characteristic

C₄×C₂₈₁

#135

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₂×C₄⋊C₈

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, direct product factor, Sylow

#1564

C₂₈₁

yes

maximal, characteristic, direct product factor, Sylow

#431

C₂×C₂₈₁

yes

—

C₂×C₄×C₈

C₂×C₂₈₁

yes

characteristic

#335

C₂×C₂₈₁

yes

complemented

#334

C₂×C₂₈₁

yes

complemented

C₂²×C₈

C₄×C₂₈₁

yes

characteristic

C₂²×C₈

C₄×C₂₈₁

yes

characteristic

C₂×C₄²

C₂²×C₂₈₁

yes

characteristic

C₄⋊C₈

C₂²×C₂₈₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×C₂₈₁

yes

characteristic

C₄×C₈

C₄×C₂₈₁

yes

characteristic

C₄×C₈

C₂²×C₂₈₁

yes

characteristic

C₄×C₈

C₂²×C₂₈₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₈₁

yes

characteristic

C₂²×C₄

D₄×C₂₈₁

yes

characteristic

C₂²×C₄

Q₈×C₂₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×C₂₈₁

yes

complemented

C₂×C₈

C₂×C₄×C₂₈₁

yes

characteristic

C₂×C₈

C₂×C₄×C₂₈₁

yes

characteristic

C₂×C₈

C₂×C₄×C₂₈₁

yes

characteristic

C₂×C₈

C₂×C₄×C₂₈₁

yes

characteristic

C₄²

C₂×C₄×C₂₈₁

yes

characteristic

C₄²

D₄×C₂₈₁

yes

characteristic

C₄²

Q₈×C₂₈₁

yes

characteristic

C₄²

C₂³×C₂₈₁

yes

characteristic

C₂³

C₄⋊C₄×C₂₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₂₈₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₂₈₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₂₈₁

yes

characteristic

C₂×C₄

C₂×C₈×C₂₈₁

yes

—

C₂×C₄

M₁₆×C₂₈₁

yes

—

C₂×C₄

C₂²×C₄×C₂₈₁

yes

characteristic

C₂×C₄

C₂×D₄×C₂₈₁

yes

characteristic

100

C₂×C₄

C₂×Q₈×C₂₈₁

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

C₂×C₈×C₂₈₁

yes

—

105

C₈

C₂×C₈×C₂₈₁

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₄⋊C₈×C₂₈₁

yes

—

109

C₂²

C₂×C₄⋊C₄×C₂₈₁

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₄⋊C₈×C₂₈₁

yes

—

113

C₄

#108×C₂₈₁

yes

—

114

C₄

C₂²×C₈×C₂₈₁

yes

characteristic

115

C₄

C₂×M₁₆×C₂₈₁

yes

characteristic

116

C₄

#135×C₂₈₁

yes

—

117

C₂

—

118

C₂

C₂×C₄⋊C₈×C₂₈₁

yes

characteristic

119

C₂

#427×C₂₈₁

yes

characteristic

120

C₂

C₂×#108×C₂₈₁

yes

characteristic

	subgroup	G/H
center	C₄²×C₂₈₁	D₄
commutator	C₄	C₂²×C₈×C₂₈₁
Frattini, cosocle	C₄²	C₂³×C₂₈₁
socle	C₂²×C₂₈₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₈₁	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₈₁ ◁ C₂×C₄²×C₂₈₁ ◁ G

subgroup

G/H

center

C₄²×C₂₈₁

D₄

commutator

C₄

C₂²×C₈×C₂₈₁

Frattini, cosocle

C₄²

C₂³×C₂₈₁

socle

C₂²×C₂₈₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₈₁

C₄⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₂₈₁ ◁ C₂×C₄²×C₂₈₁ ◁ G

N	H ≅ G/N	type
3. #335×C₂₈₁	C₂	semidirect
4. #334×C₂₈₁	C₂	semidirect
20. C₂×C₈×C₂₈₁	C₈	semidirect
60. C₂₈₁	#1564	direct (N × H)
61. #1564	C₂₈₁	direct (N × H)
64. #335	C₂×C₂₈₁	semidirect
65. #334	C₂×C₂₈₁	semidirect
81. C₂×C₈	C₈×C₂₈₁	semidirect

H ≅ G/N

type

3. #335×C₂₈₁

C₂

semidirect

4. #334×C₂₈₁

C₂

semidirect

20. C₂×C₈×C₂₈₁

C₈

semidirect

60. C₂₈₁

#1564

direct (N × H)

61. #1564

C₂₈₁

direct (N × H)

64. #335

C₂×C₂₈₁

semidirect

65. #334

C₂×C₂₈₁

semidirect

81. C₂×C₈

C₈×C₂₈₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#431×C₂₈₁	2
C₂×C₄×C₈×C₂₈₁	2
#335×C₂₈₁	2
#334×C₂₈₁	2
#1564	281

subgroup

index

#431×C₂₈₁

C₂×C₄×C₈×C₂₈₁

#335×C₂₈₁

#334×C₂₈₁

#1564

281

Subgroups — #1564×C₂₈₁ #49091089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1564×C281 #49091089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1564×C₂₈₁ #49091089